3-dimensional manifolds and related topic

摘要奖：DMS-0305846首席研究员：卡梅隆·戈登(1)该项目的目标是研究三维拓扑及其周围的几个问题。一个主要的焦点将是Cabling猜想，该猜想断言对3-球面中的双曲纽结进行Dehn运算总是产生一个质数3-流形。这是程序的一部分，以完整地描述所有双曲结上的非双曲Dehn手术。最近，John Luecke和他的主要研究人员证明了具有非积分环面Dehn手术的双曲纽结正是Eudave-Munoz所描述的，并且那里发展的一些技术可能适用于Cabling猜想。一般程序的另一部分将被解决的是这样一个猜想：在双曲线结上的任何Seifert纤维空间运算一定是积分的。该项目还将考虑与3-流形理论有关的各种几何群论主题，例如Coxeter群和Artin群的曲面子群，以及Gromov关于单端字双曲群是否总是有曲面子群的问题。另一个将被研究的关于曲面和3-流形的问题是Simple Loop猜想，它断言：如果从一个封闭的可定向曲面到封闭的可定向3-流形的映射在基本群上不是内射的，那么在该曲面上存在一个嵌入的本质圈，它的像在3-流形中是零同伦的。(2)这个项目是理解三维流形结构的一般目标的一部分。这些物体像普通的三维空间一样是局部性的，但其全球结构可能相当复杂。由于我们生活在一个三维流形中，人们可能会说，三维拓扑学的目的是描述我们的空间宇宙的数学可能性是什么。三维拓扑学的一个重要方面是节点理论--节点是以某种方式嵌入到空间中的闭合环。各种各样的数学方法可以应用到纽结的研究中，从而产生了关于三维流形的新信息。最近，人们通过这种方式发现了三维拓扑与量子物理之间的深层联系。纽结理论与三维流形的一般理论有关，通过一种被称为德恩手术的结构，在该手术中，围绕着结的实心管被移除，并以不同的方式缝合回来。我们注意到，关于Dehn手术的一个定理，循环手术定理，已经被用来确定某些酶对DNA链作用的拓扑性质。Dehn手术的许多方面现在都很好地理解了。剩下的主要问题之一，也是该项目的一个主要焦点，是证明(除了在明显退化的情况下)由Dehn手术对一个纽结产生的3-流形永远不会分解为两个更简单的流形的“和”。三维拓扑学的另一个重要工具是研究三维流形中的(二维)曲面，该项目还将解决这一领域的各种问题。最后，该项目将使首席研究员能够继续参与德克萨斯大学奥斯汀分校拓扑学研究生的教育和培训。

AbstractAward: DMS-0305846Principal Investigator: Cameron Gordon(1) The goal of the project is to investigate several problems inand around 3-dimensional topology. A major focus will be theCabling Conjecture, which asserts that Dehn surgery on ahyperbolic knot in the 3-sphere always yields a prime3-manifold. This is part of the program to completely describeall non-hyperbolic Dehn surgeries on hyperbolic knots. Recently,John Luecke and the principal investigator showed that thehyperbolic knots with non-integral toroidal Dehn surgeries areprecisely those described by Eudave-Munoz, and some of thetechniques developed there may be applicable to the CablingConjecture. Another part of the general program that will beaddressed is the conjecture that any Seifert fiber space surgeryon a hyperbolic knot must be integral. The project will alsoconsider various geometric group theoretic topics that arerelated to the theory of 3-manifolds, such as surface subgroupsof Coxeter and Artin groups, and Gromov's question as to whetheror not a 1-ended word hyperbolic group always has a surfacesubgroup. Another question concerning surfaces and 3-manifoldsthat will be investigated is the Simple Loop Conjecture, whichasserts that if a map from a closed orientable surface to aclosed orientable 3-manifold is not injective on fundamentalgroup, then there is an embedded essential loop in the surfacewhose image is null-homotopic in the 3-manifold.(2) The project is part of the general goal to understand thestructure of 3-dimensional manifolds. These are objects that arelocally like ordinary 3-dimensional space, but whose globalstructure may be quite complicated. Since we live in a3-manifold, one might say that 3-dimensional topology aims todescribe what the mathematical possibilities are for our spatialuniverse. One important aspect of 3-dimensional topology is thetheory of knots - a knot being a closed loop embedded somehow inspace. A wide variety of mathematical methods can be applied tothe study of knots, leading to new information about3-manifolds. Recently, deep connections between 3-dimensionaltopology and quantum physics were discovered in this way. Knottheory is related to the general theory of 3-manifolds through aconstruction known as Dehn surgery, in which a solid tube aroundthe knot is removed and sewn back in differently. We note that atheorem about Dehn surgery, the Cyclic Surgery Theorem, has beenused to determine the topological nature of the action of certainenzymes on strands of DNA. Many aspects of Dehn surgery are nowquite well understood. One of the main remaining questions, whichis a major focus of the project, is to show that (except in anobvious degenerate situation) the 3-manifold resulting from aDehn surgery on a knot never decomposes as a "sum" of two simplermanifolds. Another important tool in 3-dimensional topology isthe study of (2-dimensional) surfaces in 3-manifolds, and theproject will also address various questions in thisarea. Finally, the project will enable the principal investigatorto continue his involvement in the education and training ofgraduate students in topology at the University of Texas atAustin.